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MATH1061 (S1, 2023)

Assignment 1

Due: 4pm on 20 March 2023

All assignments in this course must be submitted electronically and SUBMITTED AS A SINGLE PDF FILE. Prepare your assignment solutions using Word, LaTeX, Windows Journal, or other application, ensuring that your name, student number and tutorial group number appear clearly at the top to the first page, and then save your file in pdf format.  Alternatively, you may handwrite your solutions and scan or photograph your handwritten work to create a pdf file.  Make sure that your pdf file is legible and that the file size is not excessive. Use the assignment submission link in Blackboard to submit the pdf file.

1. (3 marks) Let p, q and r be statements.  Construct a truth table for the following statement

form and determine whether it is a tautology, a contradiction, or neither of these. (欢 (p ^ (q 一欢 r)) 一 (r (p^ 欢 q))

2. (7 marks) Let p, q and r be statements.

(a) Use the Laws of Logical Equivalence and the equivalence of 一 to a disjunction to show

that:

(p (q r)) V (欢 r) 三 (p 一欢 q) V (欢 r)

Show your working and name the laws that you use at each step.

(b) A MATH1061 student reads part (a) above and remarks,

“Take a close look at the two statements we need to show are logically equivalent. Both statement include V(欢 r) at the end. So it is probably fine to just show that (p (q r)) is logically equivalent to (p 一欢 q).”

Do you agree with this student’s remark? Explain why or why not.

3. (5 marks) Determine whether the following argument is valid. Explain your reasoning.

If I go to work then I brush my teeth or drive my car.

I tie my shoes and I don’t wear socks.

If I wear socks then I don’t brush my teeth.

I go to work.

Therefore I drive my car or I don’t wear shoes.

4. (4 marks) Rewrite each of the following sentences using quantifiers (A, 3) and mathematical

notation (e, Z, V, , ...) and state whether the sentence is true or false with a brief reason.

(a) A necessary condition for a real number to be rational is that its square is rational. (b) A sufficient condition for a real number to be rational is that its square is rational.

5. (6 marks) Write down the negation of each of the following statements. Then determine whether the statement or its negation is true, and explain why.

(a) Aq e Q, 3r e Q such that e Z.

(b) 3z e Z such that Ax,y e Z, x _ y < z .

6. (5 marks) The following question appeared on an assessment item in MATH1061/7861 in a previous year.

Consider the following statement.  “There exists an integer a such that for all integers b, ab > 0.”Write down the negation of this statement. Then determine whether the statement or its negation is true, and explain why.

In February 2023, a lecturer posed this question to a generative AI model and the answer it produced is given below.   Provide a careful critique of this answer, including correcting any mistakes it may have made.

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The given  statement  is:

3a ∈ Z, Ab ∈ Z,ab > 0

The negation  of this  statement would be:

Aa ∈ Z, 3b ∈ Z,ab ≤ 0

In words, the negation  states that for  all  integers  a, there  exists  an  integer b such that  ab  is  less than  or  equal to 0 .

Now  let’s  consider whether the  original  statement  or  its negation  is true .

The  original  statement  asserts the  existence  of  an  integer  a  such that the           product  of  a  and  any  integer b  is greater than 0 . This  is true  if  a  is positive, because  in that  case, the product  of  a  and  any positive  or negative  integer will be greater than 0 . Therefore, the  original  statement  is true for  all positive     integers  a .

On the  other hand, the negation  of the  statement  asserts that for  every  integer  a, there  exists  an  integer b  such that the product  of  a  and b  is  less than  or  equal   to 0 . This  is true for  all values  of  a, because we  can  simply  choose b = 0,  and     then  ab = 0, which  is  less than  or  equal to 0 .

Therefore, the  original  statement  is true  and  its negation  is false .

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